Selected quad for the lemma: hand_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
hand_n	left_a	rank_n	right_a	50,843	5	9.3193	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64224	Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.	Taylor, John, mathematician.	1687 (1687)	Wing T534; ESTC R23734	190,995	514

the_o fractional_a part_n of_o the_o give_v mixt-number_n into_o a_o decimal_n fraction_n which_o shall_v consist_v of_o tenaries_n of_o place_n then_o to_o the_o whole_a number_n annex_v the_o decimal_n fraction_n and_o extract_v the_o cube-root_n of_o the_o whole_a and_o observe_v that_o so_o many_o point_n as_o be_v over_o the_o integer_n so_o many_o of_o the_o first_o place_n in_o the_o quotient_n must_v be_v esteem_v integer_n and_o the_o rest_n express_v the_o fractional_a part_n of_o the_o root_n in_o decimal_a part_n of_o a_o fraction_n so_o the_o cube-root_n of_o 2_o ⅜_n decimal_n 2_o 375000000_o etc._n etc._n will_v be_v find_v to_o be_v 1_o 334_o or_o 1_o 334_o 1000_o and_o be_v very_a near_o the_o true_a root_n and_o so_o for_o any_o other_o mixt-number_n of_o this_o nature_n chap._n ii_o the_o explication_n and_o use_v of_o the_o table_n of_o logarithm_n sect_n i._o the_o explication_n of_o the_o table_n of_o the_o logarithm_n and_o of_o part_n proportional_a the_o logarithm_n be_v first_o invent_v defin._n find_v out_o and_o frame_v by_o that_o never_o to_o be_v forget_v and_o thrice_o honourable_a lord_n the_o lord_n nepeir_n which_o number_n so_o find_v out_o and_o frame_v by_o his_o diligent_a industry_n he_o be_v please_v to_o call_v logarithm_n which_o in_o the_o greek_a signify_v the_o speech_n of_o number_n i_o shall_v not_o here_o trouble_v you_o with_o the_o manner_n or_o the_o construction_n of_o those_o table_n of_o logarithm_n but_o shall_v first_o lay_v down_o some_o brief_a and_o general_a rule_n that_o thereby_o the_o better_a you_o ma●_n understand_v those_o table_n and_o then_o i_o shall_v e●plain_v their_o manifold_a use_n in_o sundry_a example_n arithmetical_a etc._n etc._n prop._n i._n any_o number_n give_v under_o 10000_o or_o 100000_o to_o find_v the_o logarithm_n correspond_v thereunto_o 1._o if_o the_o number_n propound_v consist_v of_o one_o place_n who_o logarithm_n be_v require_v to_o be_v find_v as_o suppose_v 5_o look_v for_o 5_o in_o the_o top_n of_o the_o left_a hand_n column_n under_o the_o letter_n seek_v n_o and_o right_a against_o 5_o and_o in_o the_o next_o column_n under_o log_n opposite_a you_o will_v find_v this_o number_n or_o rank_n of_o figure_n 0698970_o which_o be_v the_o logarithm_n of_o the_o number_n 5_o require_v 2._o if_o the_o number_n consist_v of_o two_o place_n as_o if_o it_o be_v 57_o look_v 57_o under_o n_o and_o opposite_a to_o it_o and_o under_o log_n you_o will_v find_v this_o number_n 1._o 755875_o which_o be_v the_o logarithm_n of_o 57_o the_o number_n propound_v 3._o if_o the_o number_n propound_v consist_v of_o three_o place_n as_o 972_o look_v for_o 972_o under_o n_o and_o opposite_a to_o 972_o and_o under_o ●o_o the_o column_n you_o shall_v find_v this_o number_n 2._o 987666_o which_o be_v the_o logarithm_n of_o 972_o the_o number_n which_o be_v propound_v 4._o but_o if_o the_o number_n consist_v of_o four_o place_n as_o 685_o look_v the_o three_o first_o figure_n 168_o under_o the_o column_n n_o and_o opposite_a to_o that_o and_o un●er_v 5_o at_o the_o top_n of_o the_o page_n you_o will_v find_v this_o number_n 3._o 226599_o which_o be_v the_o logarithm_n of_o 1685_o the_o number_n propound_v 5._o but_o if_o the_o number_n book_n give_v be_v above_o 10000_o and_o under_o 100000_o you_o may_v find_v its_o logarithm_n by_o the_o table_n of_o part_n proportional_a print_v at_o the_o latter_a end_n of_o this_o book_n thus_o if_o the_o logarithm_n of_o 35786_o be_v seek_v first_o seek_v the_o log._n of_o 3578_o which_o will_v be_v 553649_o and_o the_o common_a difference_n under_o d_o be_v 121_o with_o this_o difference_n 121_o enter_v the_o table_n of_o part_n proportional_a and_o find_v 121_o in_o the_o first_o column_n under_o d_o you_o may_v then_o lineal_o under_o 6_o find_v the_o number_n 72_o which_o add_v to_o the_o log._n of_o 3578_o that_o be_v 553649_o it_o produce_v 553712_o which_o be_v the_o log._n of_o 35786_o the_o number_n propound_v now_o because_o the_o number_n propound_v 35786_o arise_v to_o the_o place_n of_o x._o m._n therefore_o there_o must_v be_v the_o figure_n 4_o prefix_v before_o its_o logarithm_n and_o then_o it_o will_v be_v thus_o 4_o 553712_o which_o 4_o be_v call_v the_o index_n as_o shall_v be_v hereafter_o show_v now_o before_o we_o proceed_v to_o find_v number_n correspond_v to_o logarithm_n it_o will_v be_v necessary_a to_o explain_v the_o meaning_n of_o the_o first_o figure_n to_o the_o left_a hand_n of_o any_o logarithm_n place_v mr._n briggs_n call_v it_o a_o characteristic_a logarithm_n or_o index_n which_o do_v represent_v the_o distance_n of_o any_o the_o first_o figure_n of_o any_o whole_a number_n from_o unity_n who_o index_n be_v 0_o a_o cypher_n so_o the_o index_n o●_n 10_o be_v 1_o and_o so_o to_o 100_o who_o index_n be_v 2_o and_o s●_n to_o 1000_o who_o index_n be_v 3_o and_o so_o to_o 10000_o who_o index_n be_v 4_o and_o so_o if_o you_o persist_v far_o the_o characteristic_a be_v always_o one_o less_o in_o dignity_n than_o the_o place_n or_o figure_n os_fw-la the_o number_n propound_v prop._n ii_o to_o find_v the_o logarithm_n belong_v to_o a_o vulgar_a fraction_n and_o a_o mix_v number_n first_o as_o be_v before_o show_v if_o it_o be_v a_o vulgar_a fraction_n find_v the_o log._n of_o the_o numerator_n and_o the_o log_n of_o the_o denominator_fw-la then_o subtract_v the_o log_n of_o the_o numerator_n from_o the_o log_n of_o the_o denominator_fw-la the_o remainder_n logarithm_n be_v the_o log_n of_o the_o fraction_n propound_v now_o if_o you_o will_v find_v the_o logarithm_n of_o 5_o 7_o do_v as_o be_v prescribe_v who_o log._n ay_o find_v to_o be_v 0._o 146121_o now_o to_o find_v the_o log._n of_o a_o mix_v number_n reduce_v it_o into_o a_o improper_a fraction_n and_o then_o do_v as_o before_o so_o the_o log_n of_o 15_o ⅖_n improper_a 77_o 5_o be_v 1_o 187_o 52_o and_o so_o do_v for_o any_o other_o mix_v number_n prop_n iii_o a_o logarithm_n propound_v to_o find_v the_o whole_a or_o mix_v number_n correspond_v thereunto_o for_o the_o more_o speedy_a find_v the_o number_n answer_v unto_o the_o logarithm_n propound_v observe_v that_o if_o the_o index_n be_v 0_o than_o the_o number_n seek_v may_v be_v find_v between_o 1_o and_o 10_o if_o 1_o between_z 10_o and_o 100_o if_o 2_o between_o 100_o and_o 1000_o if_o 3_o between_o 1000_o and_o 10000_o and_o so_o on_o still_o observe_v the_o rule_n of_o the_o characteristic_a or_o index_n therefore_o loo_n in_o the_o table_n until_o you_o find_v the_o logarithm_n propose_v and_o against_o it_o in_o the_o margin_n according_a to_o the_o aforegoing_a direction_n under_o n_o you_o shall_v find_v the_o number_n belong_v thereunto_o this_o rule_n hold_v in_o force_n in_o mix_v number_n also_o thus._n 0._o 845098_o 1._o 556302_o 2._o 130334_o 3._o 980276_o be_v the_o logarithm_n of_o 7_o 36_o 135_o 9556_o note_n but_o if_o you_o can_v find_v the_o logarithm_n exact_o in_o the_o table_n as_o in_o many_o operation_n it_o so_o happen_v you_o must_v then_o take_v the_o near_a logarithm_n number_n to_o the_o logarithm_n propound_v and_o so_o take_v the_o number_n belong_v thereto_o ●or_a the_o desire_a number_n sect_n ii_o of_o the_o admirable_a use_n of_o the_o logarithm_n in_o arithmetic_n prop._n i._o to_o multiply_v one_o number_n by_o another_o admit_v 90_o be_v to_o be_v multiply_v by_o 42_o what_o be_v the_o product_n to_o find_v which_o first_o find_v the_o log._n of_o the_o multiplicand_a 90_o who_o be_v log._n be_v 1._o 95424_o then_o find_v the_o log._n of_o logarithm_n the_o multiplier_n 42_o who_o log_n be_v 1_o 62324_o then_o add_v these_o two_o log_n together_o viz._n the_o log_n of_o the_o multiplicand_a and_o multiplier_n their_o sum_n be_v 3_o 57748_o which_o be_v the_o log_n of_o 3780_o the_o product_n of_o 90_o and_o 42_o multiply_v together_o prop._n ii_o to_o divide_v one_o number_n by_o another_o admit_v the_o dividend_n or_o number_n to_o be_v divide_v be_v 648_o and_o the_o divisor_n 72_o what_o be_v the_o number_n that_o the_o quotient_n shall_v consist_v off_o to_o find_v which_o first_o write_v down_o the_o logarithm_n of_o the_o dividend_n 648_o which_o be_v 2._o 81157_o and_o also_o write_v down_o the_o logarithm_n of_o the_o divisor_n 72_o which_o be_v 1._o logarithm_n 85733._o now_o subtract_v the_o log_n of_o the_o divisor_n out_o of_o the_o log_n of_o the_o dividend_n the_o remainder_n be_v 0._o 95424_o which_o be_v the_o logarithm_n of_o 9_o so_o i_o conclude_v that_o the_o divisor_n 72_o be_v contain_v in_o the_o dividend_n 648_o 9_o time_n and_o so_o do_v for_o any_o other_o prop._n iii_o to_o find_v the_o square-root_n of_o a_o number_n admit_v it_o be_v require_v to_o extract_n the_o square-root_n of_o the_o number_n 144_o

as_o in_o the_o former_a proposition_n prop._n iii_o in_o a_o rectangled_a triangle_n the_o base_a and_o cathetus_n give_v to_o find_v the_o other_o part_n thereof_o in_o the_o triangle_n abc_n the_o base_a ab_fw-la be_v 736_o part_n and_o the_o cathetus_n bc_n be_v 467_o 4_o 10_o part_n and_o the_o angle_n b_o between_z they_o be_v a_o right_a angle_n o●_n 90°_n and_o here_o you_o may_v make_v either_o side_n of_o the_o triangle_n radius_fw-la but_o i_o shall_v make_v bc_n the_o cathetus_n radius_fw-la and_o then_o to_o find_v the_o angle_n at_o the_o cathetus_n c_o this_o be_v the_o analogy_n or_o proportion_n as_o log._n cathet_n bc_n 467_o 4_o 10_o part_n 35._o to_z radius_fw-la or_o saint_n 90°_n so_o log._n base_n ab_fw-la 736_o part_n to_o t._n v._o cathe_n c_o 57°_n 35_o 35_o as_o require_v second_o i_o find_v the_o other_o angle_n at_o a_o to_o be_v 〈…〉_z 32°_n 25_o 25_o it_o be_v the_o compliment_n to_o c_o 57_o 35_o 35_o unto_o 90°_n three_o to_o find_v out_o the_o hypothenuse_n ac_fw-la this_o be_v the_o analogy_n or_o proportion_n as_o s._n v._o cathe_n c._n 57°_n 35_o 35_o to_o log._n base_n ab_fw-la 736_o part_n so_o radius_fw-la or_o s._n 90°_n to_o log._n hypothenuse_n ac_fw-la 871_o 8_o 10_o part_n 〈…〉_z quire_v but_o make_v the_o base_a ab_fw-la radius_fw-la yo●_n may_v find_v the_o hypothenuse_n ac_fw-la by_o this_o anal_n 〈…〉_z or_o proportion_n plate_n 1_o page_n 65_o as_o radius_fw-la or_o s._n 90°_n to_o log._n base_n ab_fw-la 736_o part_n so_o sc._n v._o base_n a_o 32°_n 25_o 25_o to_o log._n hypothenuse_n ae_n 871_o 8_o 10_o part_n require_v 35._o and_o thus_o you_o have_v all_o the_o part_n of_o the_o triangle_n propound_v prop._n iu._n the_o base_a and_o hypothenuse_n with_o the_o angle_n between_o they_o give_v to_o find_v the_o other_o part_n of_o a_o rect-angled_n triangle_n in_o the_o triangle_n abc_n the_o base_a ab_fw-la be_v 736_o part_n and_o the_o hypothenuse_n ac_fw-la be_v 871_o 8_o 10_o part_n and_o the_o angle_n a_o include_v between_o they_o be_v 32°_n 25_o 25_o first_o to_o find_v the_o angle_n and_o first_o remember_v that_o the_o angle_n b_o be_v a_o right_a angle_n or_o 90°_n second_o that_o the_o angle_n at_o c_o be_v the_o compliment_n to_o the_o angle_n at_o a_o 32°_n 25_o 25_o unto_o 90°_n 35._o and_o therefore_o be_v 57°_n 35_o 35_o now_o these_o be_v know_v you_o may_v find_v the_o cathetus_n by_o this_o analogy_n or_o proportion_n as_o s._n v._o cathe_n c._n 57°_n 35_o 35_o to_o log._n base_n ab_fw-la 736_o part_n so_o s._n v._o base_a a_o 32°_n 25_o 25_o to_o log._n cathe_n bc_n 467_o 4_o 10_o part_n require_v thus_o i_o have_v sufficient_o explain_v all_o the_o case_n of_o plain_a rect-angle_n triangle_n for_o to_o these_o rule_v they_o may_v be_v all_o reduce_v sect_n iii_o of_o obliqueangled_n plain_a triangle_n prop._n i._n two_o angle_n and_o a_o side_n opposite_a in_o a_o obliqueangled_n triangle_n give_v to_o find_v the_o other_o part_n thereof_o in_o the_o triangle_n abc_n the_o angleat_n a_o be_v 50°_n and_o at_o c_o be_v 37°_n and_o the_o side_n ab_fw-la be_v 30_o part_n and_o opposite_a to_o the_o angle_n c_o first_o to_o find_v the_o angle_n b_o remember_v that_o as_o it_o be_v say_v in_o the_o three_o maxim_n aforego_v it_o be_v the_o compliment_n to_o the_o angle_n a_o 50°_n and_o c_o 37°_n to_o 180°_n and_o therefore_o be_v the_o angle_n at_o b_o 93°_n second_o have_v thus_o find_v the_o angles_n the_o 36._o two_o unknown_a side_n may_v be_v find_v by_o the_o proportion_n they_o bear_v to_o their_o opposite_a angle_n for_o that_o proportion_n hold_v also_o in_o these_o thus_o to_o find_v the_o side_n bc_n this_o be_v the_o analogy_n or_o proportion_n as_o s._n v._n c_o 37°_n 00_o 00_o to_o log._n side_n ab_fw-la 30_o part_n so_o s._n v._o a._n 50°_n 00_o 00_o to_o log._n side_n bc_n 38_o 19_o 100_o part_n require_v to_o be_v find_v but_o it_o may_v be_v more_o ready_o find_v and_o perform_v in_o such_o case_n as_o this_o where_o you_o have_v a_o sine_fw-la or_o tangent_fw-la in_o the_o first_o place_n by_o the_o arithmetical_a compliment_n thereof_o and_o so_o save_v the_o substraction_n now_o the_o ready_a way_n to_o find_v the_o arithmetical_a number_n compliment_n be_v that_o of_o mr._n norwood_n in_o his_o doctrine_n of_o triangle_n which_o be_v thus_o begin_v with_o the_o first_o figure_n towards_o the_o left_a hand_n of_o any_o number_n and_o write_v down_o the_o compliment_n or_o the_o remainder_n thereof_o unto_o 9_o 962398._o and_o so_o do_v with_o all_o the_o 037602._o rest_n of_o the_o figure_n as_o you_o see_v here_o do_v say_v 9_o want_v of_o 9_o 0_o and_o again_o 9_o want_v 0_o 6_o want_v 3_o 2_o want_v 7_o 3_o want_v 6_o 9_o want_v 0_o only_o when_o you_o come_v to_o the_o last_o figure_n to_o the_o right_a hand_n take_v it_o out_o of_o 10_o so_z 8_o want_v 2_o of_o 10_o thus_o you_o may_v ready_o find_v the_o coat_a of_o any_o sine_fw-la almost_o as_o soon_o as_o the_o sine_fw-la itself_o but_o if_o you_o want_v the_o compliment_n arithmetical_a of_o any_o tangent_fw-la you_o may_v take_v the_o co-tang_n which_o be_v exact_o the_o co-arith_a of_o the_o double_a radius_fw-la so_o that_o the_o tangent_fw-la and_o co-tangent_a of_o a_o arch_n make_v exact_o 20._o 000000._o now_o if_o the_o radius_fw-la be_v in_o the_o first_o place_n than_o there_o be_v no_o need_n of_o take_v the_o co-arith_a of_o the_o first_o number_n only_o you_o must_v cut_v off_o the_o first_o i_o to_o the_o left_a hand_n thus_o x_o and_o you_o will_v have_v the_o logarithm_n of_o the_o number_n desire_v three_o now_o to_o find_v the_o side_n ac_fw-la by_o the_o opposite_a angle_n b_o which_o be_v 93°_n 00_o 00_o and_o see_v 〈…〉_z ng_z the_o angle_n b_o exceed_v 90°_n you_o must_v work_v 〈…〉_z y_fw-es the_o compliment_n to_o 180°_n as_o in_o the_o seven_o 〈…〉_z ork_n in_o page_n 61_o be_v teach_v thus_o have_v find_v all_o the_o part_n of_o the_o triangle_n 037602._o 36._o propound_v viz._n the_o angle_n b_o to_o be_v 93°_n 00_o 00_o the_o side_n ac_fw-la to_o be_v 49_o 78_o 100_o part_n and_o the_o side_n bc_n to_o be_v 38_o 19_o 100_o part_n as_o be_v require_v to_o be_v find_v prop._n ii_o two_o side_n and_o a_o angle_n opposite_a to_o one_o of_o they_o in_o a_o obliqueangled_n triangle_n give_v to_o find_v the_o other_o part_n thereof_o in_o the_o triangle_n abc_n the_o side_n ab_fw-la be_v 30_o part_n and_o the_o side_n ac_fw-la be_v 49_o 78_o 100_o part_n and_o the_o opposite_a angle_n c_o be_v 37°_n 00_o 00_o first_o to_o find_v the_o angle_n at_o b_o this_o be_v the_o analogy_n 36._o or_o proportion_n as_o log._n cr_n ab_fw-la 30_o part_n to_o s._n v._o at_o c_o 37°_n 00_o 00_o so_o log._n cr_n ac_fw-la 49_o 78_o 100_o part_n to_o sc._n v._n b_o 93°_n 00_o 00_o as_o be_v require_v to_o be_v find_v now_o see_v that_o the_o angle_n c_o be_v 37°_n 00_o 00_o and_o the_o angle_n b_o be_v 93°_n 00_o 00_o which_o make_v 120°_n 00_o 00_o therefore_o must_v the_o angle_v a_o be_v 50°_n 00_o 00_o the_o compliment_n to_o 180°_n so_o have_v find_v all_o the_o three_o angle_n you_o may_v find_v the_o other_o side_n cb_n 36._o 38_o 19_o 100_o part_n as_o afore_o in_o the_o first_o proposition_n by_o his_o opposite_a angle_n prop._n iii_o two_o side_n of_o a_o obliqueangled_n triangle_n with_o the_o angle_n include_v between_o they_o give_v to_o find_v the_o other_o part_n thereof_o in_o the_o triangle_n abc_n the_o side_n ac_fw-la be_v 49_o 78_o 100_o part_n the_o side_n db_n be_v 30_o part_n and_o the_o angle_n a_o between_o they_o be_v 50°_n 00_o 00_o and_o it_o be_v require_v to_o find_v the_o other_o part_n of_o the_o triangle_n propound_v to_o resolve_v this_o conclusion_n let_v fall_v a_o perpendicular_a db_n from_o the_o angle_n b_o on_o the_o side_n ac_fw-la by_o prop._n 3_o 3_o 1._o chap._n 4_o and_o then_o proceed_v thus_o first_o see_v the_o obliqueangled_n triangle_n 37._o abc_n be_v divide_v into_o two_o rectangled_a triangle_n viz._n adb_n and_o bdc_a now_o i_o will_v begin_v with_o the_o triangle_n adb_n in_o which_o be_v give_v the_o angle_n a_o 50°_n 00_o 00_o and_o the_o angle_n d_o be_v a_o right_a angle_n or_o 90°_n and_o the_o side_n ab_fw-la 30_o part_n and_o the_o side_n ad_fw-la and_o db_n and_o the_o angle_n at_o b_o be_v require_v first_o to_o find_v the_o angle_n at_o b_o remember_v that_o it_o be_v the_o compliment_n unto_o the_o angle_n a_o 50°_n 00_o 00_o unto_o 90°_n 00_o 00_o and_o therefore_o must_v the_o angle_n b_o be_v 40°_n 00_o 00_o now_o for_o to_o find_v the_o cathetus_n bd_o as_o in_o prop._n 1._o and_o 2_o 2_o

2._o chap._n 5._o by_o the_o rule_n of_o opposition_n the_o analogy_n or_o proportion_n hold_v thus_o as_o radius_fw-la or_o s._n 90°_n to_o log._n hypoth_o ab_fw-la 30_o part_n so_o s._n v._o at_o a_o 50°_n 00_o 00_o to_o log._n cath._n bd_o 22_o 98_o 100_o part_n seek_v and_o again_o say_v as_o radius_fw-la or_o s._n 90°_n to_o log._n hypoth_o ab_fw-la 30_o part_n so_o s._n v._o at_o b_o 40°_n 00_o 00_o to_o log._n base_n ad_fw-la 19_o 28_o 100_o part_n seek_v thus_o in_o the_o triangle_n adb_n you_o have_v find_v the_o angle_n b_o to_o be_v 40°_n 00_o 00_o the_fw-fr cathetus_n bd_o to_o be_v 22_o 98_o 100_o part_n and_o the_o base_a ad_fw-la to_o be_v 19_o 28_o 100_o part_n as_o be_v so_o require_v now_o for_o the_o other_o triangle_n which_o be_v bdc_a in_o which_o there_o be_v give_v the_o side_n bd_o 22_o 98_o 100_o part_n and_o the_o angle_n at_o d_o be_v a_o right-angle_n or_o 90°_n and_o the_o side_n dc_o and_o cb_n and_o the_o angle_n b_o and_o c_o be_v require_v first_o to_o find_v the_o side_n dc_o subtract_v ad_fw-la 19_o 28_o 100_o 37._o part_n out_o of_o ac_fw-la 49_o 78_o 100_o part_n there_o remain_v the_o base_a dc_o 30_o 50_o 100_o part_n thus_o have_v you_o the_o two_o side_n of_o the_o triangle_n to_o wit_v the_o base_a dc_o 30_o 50_o 100_o part_n and_o the_o cathetus_n bd_o 22_o 98_o 100_o part_n and_o the_o angle_n d_o between_o they_o be_v a_o right-angle_n or_o 90°_n now_o you_o may_v find_v the_o angle_n at_o b_o by_o the_o tangent_fw-la as_o in_o prop._n 3_o 3_o 2._o chap._n 5._o thus_o as_o log._n cath._n bd_o 22_o 98_o 100_o part_n to_z radius_fw-la or_o s._n 90°_n so_o log._n base_a cd_o 30_o 50_o 100_o part_n to_o t._n v._o b._n 53°_n 00_o 00_o second_o for_o the_o angle_n c_o remember_v it_o be_v the_o compliment_n of_o the_o angle_n b_o 53°_n to_o 90°_n and_o therefore_o be_v the_o angle_n c_o 37°_n 00_o 00_o require_v three_o to_o find_v the_o hypoth_n bc_n this_o be_v the_o analogy_n or_o proportion_n as_o s._n v._o b._n 53°_n 00_o 00_o to_o log._n base_a dc_o 30_o 50_o 100_o part_n so_o radius_fw-la or_o s._n 90°_n to_o log._n hypoth_o bc_n 38_o 19_o 100_o part_n thus_o have_v you_o find_v all_o the_o require_v part_n of_o the_o triangle_n abc_n propound_v viz._n the_o angle_n c_o to_o be_v 37°_n 00_o 00_o the_o angle_n b_o to_o be_v 93°_n 00_o 00_o proposition_n and_o the_o side_n bc_n 38_o 19_o 10●_o part_n as_o require_v to_o be_v find_v another_o way_n to_o perform_v the_o same_o take_v the_o sum_n of_o the_o proposition_n two_o side_n and_o the_o difference_n of_o the_o two_o side_n and_o work_v as_o follow_v now_o to_o find_v the_o two_o 37._o angle_n b_o and_o c_o this_o be_v the_o manner_n and_o by_o this_o analogy_n or_o proportion_n they_o be_v find_v out_o and_o know_v as_o log._n z._n cr_n s._n ab_fw-la and_o ca_n 79_o 78_o 100_o part_n to_o log._n x._o cr_n s._n ab_fw-la and_o ca_n 19_o 78_o 100_o part_n so_o t._n of_o ½_o w_n unknown_a 65°_n 00_o 00_o to_o t._n ½x_n of_o w_n 28°_n 00_o 00_o this_o difference_n of_o angel_n 28°_n 00_o 00_o add_v unto_o 65°_n 00_o 00_o half_o the_o difference_n of_o the_o unknown_a angle_n and_o it_o shall_v produce_v 93°_n 00_o 00_o which_o be_v the_o great_a angle_n and_o substract_v from_o it_o leave_v 37°_n 00_o 00_o which_o be_v the_o lesser_a angle_n c_o so_o have_v you_o the_o require_v angle_n prop._n iu._n the_o three_o side_n of_o a_o obliqueangled_n triangle_n give_v to_o find_v the_o angles_n in_o the_o triangle_n abc_n the_o side_n ac_fw-la be_v 49_o 78_o 100_o part_n the_o side_n ab_fw-la be_v 30_o part_n and_o the_o side_n bc_n be_v 38_o 19_o 100_o part_n and_o the_o three_o angle_n of_o the_o triangle_n be_v require_v the_o resolution_n of_o 37._o this_o conclusion_n be_v thus_o take_v the_o sum_n and_o differ_v 37._o of_o the_o two_o side_n ab_fw-la and_o bc_n and_o then_o work_v as_o follow_v to_o find_v a_o segment_n of_o the_o base_a ac_fw-la to_o wit_n ce_fw-fr say_v as_o log._n base_n ac_fw-la 49_o 78_o 100_o part_n to_o z._n cr_n s._n ab_fw-la and_o bc_n 68_o 19_o 100_o part_n so_o x._o cr_n s._n ab_fw-la and_o bc_n 8_o 19_o 100_o part_n to_o log_n of_o a_o segment_n of_o the_o base_a ac_fw-la to_o wit_n c_o e_o 11_o 22_o 100_o part_n this_o segment_n of_o the_o base_a ce_fw-fr 11_o 22_o 100_o part_n be_v substract_v from_o the_o whole_a base_a ac_fw-la 49_o 78_o 100_o part_n the_o remainder_n be_v ea_fw-la 38_o 56_o 100_o part_n in_o the_o middle_n of_o which_o as_o at_o d_o the_o perpendicular_a db_n will_v fall_v from_o the_o angle_n b_o and_o so_o divide_v 37._o it_o into_o two_o rectangled_a triangle_n to_o wit_n adb_n and_o cdb_n who_o base_a da_fw-la be_v 19_o 28_o 100_o part_n which_o take_v from_o ac_fw-la 49_o 78_o 100_o part_n leave_v the_o base_a of_o the_o great_a triangle_n cd_o 30_o 50_o 100_o part_n now_o have_v the_o two_o base_n of_o these_o two_o triangle_n and_o their_o hypothenuse_n to_o wit_n cd_o 30_o 50_o 100_o part_n da_z 19_o 28_o 100_o part_n cb_n 38_o 19_o 100_o part_n and_o ba_z 30_o part_n you_o may_v find_v all_o their_o angle_n by_o the_o rule_n of_o opposite_a side_n to_o their_o angle_n as_o afore_o i._o in_o the_o triangle_n cdb_n to_o find_v the_o angle_n this_o be_v the_o analogy_n or_o proportion_n as_o log._n bc_n 38_o 19_o 100_o part_n to_z radius_fw-la or_o s._n 90°_n so_o log._n dc_o 30_o 50_o 100_o part_n to_o s._n v._n b_o 53°_n 00_o 00_o who_o compliment_n be_v the_o 37._o angle_n at_o c_o 37°_n 00_o 00_o unto_z 90_o or_o a_o quadrant_n ii_o in_o the_o triangle_n adb_n to_o find_v the_o angle_n this_o be_v the_o analogy_n or_o proportion_n as_o log._n ab_fw-la 30_o part_n to_z radius_fw-la or_o s._n 90°_n so_o log._n ad_fw-la 19_o 28_o 100_o part_n to_o s._n v._n b_o 40°_n 00_o 00_o the_o compliment_n whereof_o unto_o 90°_n 00_o 00_o be_v the_o angle_n at_o a_o 50°_n 00_o 00_o now_o in_o the_o first_o triangle_n cdb_n there_o be_v find_v the_o angle_n c_o to_o be_v 37°_n 00_o 00_o and_o the_o angle_n b_o to_o be_v 53°_n 00_o 00_o in_o the_o second_o triangle_n adb_n there_o be_v find_v the_o angle_n a_o to_o be_v 50°_n 00_o 00_o and_o the_o angle_n b_o to_o be_v 40°_n 00_o 00_o now_o the_o two_o angle_n at_o b_o to_o wit_n 53°_n 00_o 00_o and_o 40°_n 00_o 00_o make_v 93°_n 00_o 00_o which_o be_v the_o angle_n of_o the_o obliqueangled_n triangle_n abc_n at_o b_o thus_o the_o three_o angle_n of_o the_o say_v give_v triangle_n abc_n be_v find_v as_o be_v require_v viz._n the_o angle_n a_o to_o be_v 50°_n 00_o 00_o the_o angle_n b_o to_o be_v 93°_n 00_o 00_o and_o the_o angle_n c_o to_o be_v 37°_n 00_o 00_o as_o seek_v thus_o i_o have_v sufficient_o full_o and_o plain_o explain_v all_o the_o case_n of_o plain_a rightlined_n triangle_n both_o right_a and_o obliqueangled_n i_o shall_v now_o fall_v in_o hand_n with_o spherical_a triangle_n both_o right_a and_o obliqueangled_n sect_n iv._o of_o spherical_a rectangled_a triangle_n and_o here_o first_o it_o will_v be_v necessary_a also_o to_o understand_v those_o few_o general_a maxim_n or_o rule_n that_o be_v of_o special_a moment_n in_o the_o doctrine_n of_o spherical_a triangle_n 1._o that_o a_o spherical_a triangle_n be_v comprehend_v and_o form_v by_o the_o conjunction_n and_o intersection_n of_o three_o arch_n of_o a_o circle_n describe_v on_o the_o surface_n of_o the_o sphere_n or_o globe_n 2._o that_o those_o spherical_a triangle_n consist_v of_o six_o distinct_a part_n viz._n three_o side_n and_o three_o angle_n any_o of_o which_o be_v know_v the_o other_o be_v also_o find_v out_o and_o know_v 3._o that_o the_o three_o side_n of_o a_o spherical_a triangle_n be_v part_n or_o arch_n of_o three_o great_a circle_n of_o a_o sphere_n mutual_o intersection_n each_o other_o and_o as_o plain_a or_o rightlined_n triangle_n be_v measure_v by_o a_o measure_n or_o scale_n of_o equal_a part_n so_o these_o be_v measure_v by_o a_o scale_n or_o arch_n of_o equal_a degree_n 4._o that_o a_o great_a circle_n be_v such_o a_o circle_n that_o do_v bessect_v the_o sphere_n divide_v it_o into_o two_o equal_a part_n as_o the_o equinoctial_a the_o ecliptic_a the_o meridian_n the_o horizon_n etc._n etc._n 5._o that_o in_o a_o rightangled_n spherical_a triangle_n the_o side_n subtend_v the_o right-angle_n we_o call_v the_o hypothenuse_n the_o other_o two_o contain_v the_o right-angle_n we_o may_v simple_o call_v the_o side_n and_o for_o distinction_n either_o of_o they_o may_v be_v call_v the_o base_a or_o perpendicular_a 6._o that_o the_o sum_n of_o the_o

edge_n shall_v show_v the_o hour_n of_o the_o day_n now_o to_o draw_v the_o hour-line_n with_o the_o radius_fw-la of_o your_o line_n of_o chord_n on_o m_n strike_v the_o arch_a qn_n which_o divide_v into_o 5_o equal_a part_n in_o the_o point_n ●_o ●_o ●_o etc._n etc._n then_o lay_v a_o ruler_n from_o m_n unto_o each_o of_o those_o point_n and_o it_o will_v cut_v the_o line_n jk_n in_o the_o point_n point_n point_n point_n etc._n etc._n through_o which_o point_n by_o prop._n 4._o 4._o 1._o chap._n 4._o draw_v parallel_n to_o 6_o i_o 6_o as_o the_o line_n 77_o 88_o etc._n etc._n which_o shall_v be_v the_o true_a hour-line_n of_o a_o east_n plane_n from_o 6_o 66._o in_o the_o morning_n till_o 11_o before_o noon_n then_o for_o the_o hour-line_n of_o 4_o and_o 5_o you_o must_v prick_v off_o 5_o as_o far_o from_o 6_o as_o 6_o be_v from_o 7_o and_o 4_o as_o far_z as_o 6_o be_v from_o 8_o and_o draw_v the_o hour-line_n 55_o and_o 44_o as_o before_o thus_o be_v your_o dial_n complete_v and_o in_o the_o form_n of_o which_o you_o have_v make_v both_o a_o east_n and_o a_o west_n dial_n which_o be_v the_o same_o in_o all_o respect_n only_o whereas_o the_o arch_n h_n g_o through_o which_o the_o equinoctial_a pass_v in_o the_o east_n dial_n be_v describe_v on_o the_o right_a hand_n of_o the_o plane_n in_o the_o west_n it_o must_v be_v draw_v on_o the_o left_a hand_n and_o the_o hour-line_n 4_o 5_o 6_o 7_o 8_o 9_o 10_o and_o 11_o in_o the_o forenoon_n in_o the_o east_n dial_n must_v be_v 8_o 7_o 6_o 5_o 4_o 3_o 2_o and_o 1_o in_o the_o west_n in_o 67._o the_o afternoon_n as_o in_o the_o figure_n plain_o appear_v now_o you_o may_v find_v the_o distance_n of_o the_o hour-line_n from_o the_o substile_a by_o this_o analogy_n or_o proportion_n as_o the_o radius_fw-la to_o the_o height_n of_o the_o style_n so_o be_v the_o tangent_fw-la of_o any_o hour_n distance_n from_o 6_o to_o the_o distance_n thereof_o from_o the_o substile_a prop._n iu._n how_o to_o draw_v the_o hour-line_n on_o a_o direct_a south_n and_o north_n plane_n this_o plane_n or_o dial_n must_v stand_v upright_o have_v his_o face_n or_o plane_n if_o it_o be_v a_o south_n dial_n direct_o opposite_a unto_o the_o south_n but_o if_o a_o north_n plane_n direct_o opposite_a unto_o the_o north_n now_o admit_v it_o be_v require_v to_o make_v a_o direct_a south_n dial_n 68_o for_o the_o latitude_n of_o 51°_n 32_o 32_o to_o make_v which_o first_o describe_v the_o circle_n abcd_n to_o represent_v a_o e●ect_n direct_a south_n plane_n cross_v it_o with_o the_o diameter_n cb_n and_o ad_fw-la then_o out_o of_o your_o line_n of_o chord_n take_v 38°_n 28_o 28_o the_o compliment_n of_o the_o latitude_n and_o set_v it_o from_o a_o unto_o a_o and_o from_o b_o unto_z b_o then_o lay_v a_o ruler_n from_o c_o unto_o a_o and_o it_o will_v cut_v the_o meridian_n ard_n in_o p_o the_o pole_n of_o plate_n five_o page_n 261_o as_o radius_fw-la or_o s._n 90°_n to_o sc._n of_o the_o latitude_n so_o be_v t._n of_o the_o hour_n from_o noon_n to_o t._n of_o the_o hour-line_n from_o the_o meridian_n prop._n v._o how_o to_o draw_v the_o hour-line_n on_o a_o horizontal_n plane_n this_o horizontal_n plane_n or_o dial_n be_v one_o of_o the_o best_a and_o most_o useful_a dial_n in_o our_o oblique_a hemisphere_n admit_v it_o be_v require_v to_o make_v a_o horizontal_n dial_n for_o the_o latitude_n of_o 51°_n 32_o 32_o to_o make_v which_o first_o describe_v the_o circle_n ab_fw-la 70._o cd_o which_o represent_v your_o horizontal_n plane_n then_o cross_v it_o with_o the_o two_o diameter_n arc_n and_o brd_n then_o take_v 51°_n 32_o 32_o out_z of_o your_o line_n of_o chord_n and_o set_v it_o from_o b_o to_z a_o and_o from_o c_o to_z b_o then_o lay_v a_o ruler_n from_o a_o unto_o a_o and_o it_o will_v cut_v the_o meridian_n bd_o in_o p_o the_o pole_n of_o the_o world_n then_o lay_v a_o ruler_n from_o a_o unto_z b_o and_o it_o will_v cut_v abdella_n the_o meridian_n in_o the_o point_n ae_n where_o the_o equinoctial_a cut_v the_o meridian_n then_o through_o the_o three_o point_n a_o ae_n and_o c_o draw_v 70._o the_o equinoctial_a circle_n who_o centre_n be_v at_o h_n and_o find_v as_o in_o the_o former_a proposition_n then_o divide_v the_o semicircle_n adc_n into_o 12_o equal_a part_n in_o the_o point_n ●_o ●_o ●_o etc._n etc._n then_o lay_v a_o ruler_n to_o r_o the_o centre_n of_o the_o plane_n and_o on_o those_o point_n so_o shall_v the_o equinoctial_a circle_n aaec_fw-la be_v by_o it_o divide_v into_o 12_o unequal_a part_n in_o the_o point_n point_n point_n point_n point_n etc._n etc._n then_o a_o ruler_n lay_v unto_o p_o the_o pole_n of_o the_o world_n and_o those_o point_n shall_v cut_v the_o semicircle_n cda_n in_o those_o point_n i_o i_o i_o etc._n etc._n last_o from_o the_o centre_n r_o and_o through_o those_o point_n let_v there_o be_v draw_v right_a line_n which_o shall_v be_v the_o true_a hour-line_n of_o such_o a_o horizontal_n plane_n from_o 6_o in_o the_o morning_n until_o 6_o at_o night_n but_o for_o the_o hour_n of_o 4_o and_o 5_o in_o 70._o the_o morning_n and_o 7_o and_o 8_o in_o the_o evening_n they_o be_v delineate_v by_o produce_v 4_o and_o 5_o in_o the_o evening_n through_o the_o centre_n r_o and_o 7_o and_o 8_o in_o the_o morning_n extend_v they_o out_o unto_o the_o other_o side_n of_o the_o plane_n so_o shall_v you_o have_v those_o hour-line_n also_o on_o your_o plane_n delineate_v as_o you_o see_v in_o the_o figure_n the_o style_n of_o this_o plane_n may_v be_v a_o thin_a plate_n of_o brass_n cut_v exact_o unto_o the_o quantity_n of_o a_o angle_n of_o 51°_n 32_o 32_o and_o set_v perpendicular_a on_o the_o meridian_n line_n for_o the_o form_n of_o this_o style_n take_v out_o of_o your_o line_n of_o chord_n 51°_n 32_o 32_o and_o set_v it_o from_o d_o unto_z e_z and_o draw_z re_z which_o shall_v be_v the_o axis_n of_o the_o style_n you_o may_v also_o prefix_v the_o half_n and_o quarter_n of_o hour_n in_o the_o very_a same_o manner_n as_o the_o hour_n themselves_o be_v draw_v now_o to_o find_v out_o the_o distance_n of_o the_o hour-line_n from_o the_o meridian_n say_v as_o the_o radius_fw-la or_o s._n 90°_n to_o the_o s._n of_o the_o latitude_n so_o be_v the_o t._n of_o the_o hour_n from_o noon_n to_o the_o t._n of_o the_o hour-line_n from_o the_o meridian_n line_n these_o kind_n of_o dial_n be_v so_o frequent_o use_v 70._o with_o we_o in_o this_o oblique_a sphere_n for_o the_o help_n of_o the_o speedy_a delineat_n of_o they_o i_o have_v annex_v hereunto_o the_o table_n of_o longomontanus_n wherein_o the_o hour-line_n for_o many_o latitude_n be_v calculate_v a_o table_n show_v the_o distance_n of_o the_o hour-line_n from_o the_o meridian_n in_o these_o degree_n of_o latitude_n a_o horizontal_n dial_n latitude_n the_o hour_n from_o the_o meridian_n a_o south_n erect_v dial_n latitude_n xi_o i._o x._o two_o ix_o three_o viij_o four_o seven_o v._n vi_fw-la d_o m_o d_o m_o d_o m_o d_o m_o d_o m_o d_o m_o 30_o 7_o 38_o 16_o 6_o 26_o 34_o 40_o 54_o 61_o 49_o 90_o 00_o 60_o 31_o 7_o 51_o 16_o 34_o 27_o 14_o 41_o 42_o 62_o 28_o 90_o 00_o 59_o 32_o 8_o 4_o 17_o 1_o 27_o 53_o 42_o 30_o 63_o 6_o 90_o 00_o 58_o 33_o 8_o 17_o 17_o 27_o 28_o 34_o 43_o 17_o 63_o 45_o 90_o 00_o 57_o 34_o 8_o 30_o 17_o 54_o 29_o 13_o 44_o 5_o 64_o 42_o 90_o 00_o 56_o 35_o 8_o 43_o 18_o 20_o 29_o 49_o 44_o 46_o 64_o 56_o 90_o 00_o 55_o 36_o 8_o 56_o 18_o 45_o 30_o 25_o 45_o 28_o 65_o 27_o 90_o 00_o 54_o 37_o 9_o 9_o 19_o 9_o 31_o 1_o 46_o 9_o 65_o 58_o 90_o 00_o 53_o 38_o 9_o 21_o 19_o 34_o 31_o 37_o 46_o 50_o 66_o 29_o 90_o 00_o 52_o 39_o 9_o 33_o 19_o 57_o 32_o 9_o 47_o 26_o 66_o 55_o 90_o 00_o 51_o 40_o 9_o 46_o 20_o 20_o 32_o 40_o 48_o 1_o 67_o 20_o 90_o 00_o 50_o 41_o 9_o 58_o 20_o 43_o 33_o 14_o 48_o 37_o 67_o 45_o 90_o 00_o 49_o 42_o 10_o 10_o 21_o 7_o 33_o 47_o 49_o 13_o 68_o 11_o 90_o 00_o 48_o 43_o 10_o 22_o 21_o 29_o 34_o 17_o 49_o 44_o 68_o 32_o 90_o 00_o 47_o 44_o 10_o 24_o 21_o 50_o 34_o 46_o 50_o 14_o 68_o 52_o 90_o 00_o 46_o 45_o 10_o 43_o 22_o 12_o 35_o 15_o 50_o 45_o 69_o 14_o 90_o 00_o 45_o 46_o 10_o 54_o 22_o 33_o 35_o 44_o 51_o 16_o 69_o 37_o 90_o 00_o 44_o 47_o 11_o 5_o 22_o 33_o 36_o 10_o 51_o 43_o 69_o 53_o 90_o 00_o 43_o 48_o 11_o 16_o 23_o 12_o 36_o 35_o 52_o 9_o 70_o 10_o 90_o 00_o 42_o 49_o 11_o 26_o 23_o 32_o 37_o 1_o 52_o 35_o 70_o 28_o